The vertex conditions described in the corollary are called scale-invariant, because they do not couple the function values \(\phi (a)\)  and the values of derivatives \(\phi ^{\prime }(a)\) . Then the operator \(\mathcal {H}\)  has no eigenvalues, and its continuous spectrum coincides with \([0,+\infty )\) . There are also other properties of the operators with the scale-invariant vertex couplings, such as the factorization
of quantum graph Hamiltonians and the energy independence of vertex scattering matrices [1]}.
